real analysis - A necessary condition to $F'(x)=f(x)$ for a continuous function $f$

Theorem: Consider ,

$$F(x)=\int_a^xf(t)\,dt$$

If the function $f:[a,b]\to \mathbb R$ is continuous then , $F(x)$ is differentiable and $F'(x)=f(x).$

I know that the continuity condition of $f$ is sufficient condition.

That means there exists a discontinuous function $f$ for which this $F'(x)=f(x)$.

My Question:

Does there exist a necessary condition for this ?

$$OR$$

After imposing which extra condition on $f$ it is necessary that $F'(x)=f(x)$ ?